Second Neighborhood via First Neighborhood in Digraphs

Abstract.

Let D be a simple digraph without loops or digons. For any \( v\in V(D) \), the first out-neighborhood N+(v) is the set of all vertices with out-distance 1 from v and the second neighborhood N++(v) of v is the set of all vertices with out-distance 2 from v. We show that every simple digraph without loops or digons contains a vertex v such that \( |N^{++}(v)|\geq\gamma|N^+(v)| \), where γ = 0.657298... is the unique real root of the equation 2x3 + x2 -1 = 0.